UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA doi: 10.4467/20843828AM.16.005.5377 53(2016), 39{57 ALGEBRA OF OPERATORS AFFILIATED WITH A FINITE TYPE I VON NEUMANN ALGEBRA by Piotr Niemiec and Adam Wegert Abstract. The aim of the paper is to prove that the ∗-algebra of all (closed densely defined linear) operators affiliated with a finite type I von Neumann algebra admits a unique center-valued trace, which turns out to be, in a sense, normal. It is also demonstrated that for no other von Neumann algebras similar constructions can be performed. 1. Introduction. With every von Neumann algebra A one can associate the set Aff(A) of operators (unbounded, in general) which are affiliated with A. In [11] Murray and von Neumann discovered that, surprisingly, Aff(A) turns out to be a unital ∗-algebra when A is finite. This was in fact the first example of a rich set of unbounded operators in which one can define algebraic binary op- erations in a natural manner. This concept was later adapted by Segal [17,18], who distinguished a certain class of unbounded operators (namely, measurable with respect to a fixed normal faithful semi-finite trace) affiliated with an ar- bitrary semi-finite von Neumann algebra and equipped it with a structure of a ∗-algebra (for an alternative proof see e.g. [12] or x2 in Chapter IX of [21]). A more detailed investigations in algebras of the form Aff(A) were initiated by a work of Stone [19], who described their models for commutative A in terms of unbounded continuous functions densely defined on the Gelfand spectrum X of A. Much later Kadison [6] studied this one-to-one correspondence between operators in Aff(A) and functions on X. Recently Liu [9] established an in- teresting property of Aff(A) concerning the Heisenberg uncertainty principle. 2010 Mathematics Subject Classification. Primary 46L10; Secondary 47C15. Key words and phrases. Finite type I von Neumann algebra; operator affiliated with a von Neumann algebra; center-valued trace. The first author is supported by the Polish National Science Centre (NCN), Decision No. DEC-2013/11/B/ST1/03613. 40 Namely, she showed that the canonical commutation relation, which has the form AB − BA = I, fails to hold for any A; B 2 Aff(A), provided A is finite. For finite type I algebras this result is a simple corollary of a fact, more or less known to the experts, that Aff(A) has a uniquely determined center-valued trace, provided A is a finite type I von Neumann algebra. In this paper we give a proof of this fact. Our result reads as follows: Theorem 1.1. Let A be a finite type I von Neumann algebra and let Aff(A) be the ∗-algebra of all operators affiliated with A. Then there is a unique linear map trAff : Aff(A) ! Z(Aff(A)) such that (tr1) trAff(A) is non-negative, provided A 2 Aff(A) is so; (tr2) trAff(X · Y ) = trAff(Y · X) for any X; Y 2 Aff(A); (tr3) trAff(Z) = Z for each Z 2 Z(Aff(A)). Moreover, (1-1) Z(Aff(A)) = Aff(Z(A)) and (tr4) trAff(A) 6= 0 provided A 2 Aff(A) is non-zero and non-negative; (tr5) trAff(X · Z) = trAff(X) · Z for any X 2 Aff(A) and Z 2 Z(Aff(A)); (tr6) every increasing net (Aσ)σ2Σ of self-adjoint members of Aff(A) which is majorized by a self-adjoint operator in Aff(A) has its least upper bound in Aff(A), and (1-2) sup trAff(Aσ) = trAff sup Aσ : σ2Σ σ2Σ The above result, in a little bit different settings, was earlier established by Berberian [1], who used totally different methods. Besides, we establish more properties of the trace than he did. All they (as well as (1-1), which holds for all finite von Neumann algebras A and most of the content of Sections 2 and 3) are, however, more or less known to the experts and can be deduced from the possibility of modelling Aff(A) as a direct sum of finite-dimensional matrix algebras over corresponding algebras of bounded measurable functions (cf. [15] for a similar result in a more general setting). Nevertheless, it is likely that this result nowhere appears explicitly. For the reader's convenience, we prove most of auxiliary results here. It is worth noting that (1-2) is a natural counterpart of normality (in the terminology of Takesaki { see Definition 2.1 in Chapter V of [20]) of center- valued traces in finite von Neumann algebras. The existence of the l.u.b. in item (tr6) was established by Yeadon [22] for more general algebras than Aff(A). It is natural to ask whether the above result may be generalised to a wider class of von Neumann algebras (e.g. for all finite ones). Our second goal is 41 to show that the answer is negative, which is somewhat surprising. A precise formulation of the result is stated below. We recall that, in general, the set Aff(A) admits no structure of a vector space, nevertheless, it is always ho- mogeneous and for any T 2 Aff(A) and S 2 A the operator T + S is a (well defined) member of Aff(A). Based on this observation, we may formulate our result, which appears to be new, as follows. Proposition 1.2. Let A be a von Neumann algebra and let Aff(A) be the set of all operators affiliated with A. Assume there exists a function ': Aff(A) ! Aff(A) with the following properties: (a) if A; B 2 A are such that '(A) 2 A, then '(αA + βB) = α'(A) + β'(B) for any scalars α; β 2 C; (b) '(A) is non-negative, provided A 2 Aff(A) is so; (c) if A 2 Aff(A) and B 2 A are non-negative, and '(B) 2 A, then '(A+B) = '(A) + '(B); (d) '(AB) = '(BA) for all A; B 2 A; (e) '(Z) = Z for each Z 2 Z(A); (f) if A and U are members of A and U is unitary, then U ∗'(A)U = '(A). Then A is finite and type I. The paper is organized as follows. In the next section we establish an inter- esting property of finite type I von Neumann algebras, which is crucial in this paper, since all other results, apart from Proposition 1.2, are its consequences. Its proof involves measure-theoretic techniques, which is in contrast to all other parts of the paper, where all arguments are, roughly speaking, intrinsic and algebraic. In Section 3 we establish most relevant properties of the set Aff(A) (for a finite type I algebra A), including a new proof of the fact that Aff(A) admits a structure of a ∗-algebra. In Section 4 we introduce the center-valued trace on Aff(A) and prove all items of Theorem 1.1, except for (tr6), which is shown in Section 5, where we also establish other order properties of Aff(A). Finally, Section 6 contains a proof of Proposition 1.2. Notation and terminology. In this paper A is used to denote an arbitrary von Neumann algebra acting on a (complex) Hilbert space H. All operators are linear, closed and densely defined in a Hilbert space, projections are orthogonal while non-negative operators are, by definition, self-adjoint. The algebra of all bounded operators on H is denoted by B(H). An operator T in H is affiliated with A if UTU −1 = T for any unitary operator U belonging to the commutant A0 of A. Aff(A) stands for the set of all operators affiliated with A. If A is finite, Aff(A) may be naturally equipped with the structure of a ∗-algebra (see e.g. [11]). In that case we denote binary algebraic operations in Aff(A) by `+++' (for addition), `−' (for subtraction) and `·' (for multiplication). For any ring R, Z(R) stands for the center of R (that is, Z(R) consists of all elements of R 42 which commute with any element of R). This mainly applies to R = A, and R = Aff(A), provided A is finite. For any operator S, we use D(S), N (S) and R(S) to denote the domain, kernel and range of S, respectively. By jSj we ∗ 1 L denote the operator (S S) 2 . For any collection fTsgs2S of operators, s2S Ts is understood as an operator with a maximal domain defined naturally; that L is, ⊕s2Sxs belongs to the domain of s2S Ts if xs 2 D(Ts) for each s 2 S, and P 2 L s2S kTsxsk < 1 (and, of course, ( s2S Ts)(⊕s2Sxs) = ⊕s2S(Tsxs)). The center-valued trace in a finite von Neumann algebra W is denoted by trW. The center-valued trace on the algebra of operators affiliated with a finite type I von Neumann algebra will be denoted by trAff. All vector spaces are assumed ∗ to be over the field C of complex numbers. For two C -algebras C1 and C2, ∼ we write C1 = C2 when C1 and C2 are ∗-isomorphic. The direct product of ∗ Q a collection fCsgs2S of C -algebras is denoted by s2S Cs and it consists of all systems (as)s2S with as 2 Cs and sups2S kask < 1 (cf. Definition II.8.1.2 in [2]). By I we denote the identity operator on H. 2. Key result. As we will see in the sequel, all our main results depend on the following theorem, whose proof is the purpose of this section.